An infinite family of Hadamard matrices constructed from Paley type matrices

Abstract

An $n\times n$ matrix whose entries are from the set $\{1,-1\}$ is called a Hadamard matrix if $HH^{\top} = nI_n$. The Hadamard conjecture states that if $n$ is a multiple of four then there always exists Hadamard matrices of this order. But their construction remains unknown for many orders. In this paper we construct Hadamard matrices of order $2q(q+1)$ from known Hadamard matrices of order $2(q+1)$, where $q$ is a power of a prime number congruent to $1$ modulo $4$. We show then two ways to construct them. This work is a continuation of U. Scarpis' in \cite{c5} and Dragomir-\u{Z}. Dokovi\`{c}'s in \cite{c8}.